23. Let AB represent the fissure, AU, BV, the physical lines perpendicular to it, and passing close to its extremities, and for the greater distinctness, let us suppose the boundary of the lamina along UV to be parallel to the fissure. Let EF be a physical line originally parallel to the straight line AB. After the formation of the fissure,

it will evidently assume a curved form resembling that of APB; but its curvature will be less than that of the latter line, since the curvature of all such lines must obviously be smaller the nearer they are situated to the fixed straight line UV, along which it becomes evanescent. If, however, the length of the fissure be considerable, the curvature of APB will be very small, and therefore the variation of curvature in successive physical lines such as EF, will be extremely slow, AU being very large *.

Also, let PQS be a physical line, parallel before the opening of the fissure, to AU. If the form of every such line as EQF were exactly the same, this line would still be accurately straight and parallel to AU, and consequently in the case we are supposing it will be approximately so. The tension of all such lines will evidently be much affected by the opening of the fissure. Since there is no force acting at P, the tension of SP in the direction of its length, will, at that

*

If the boundary of the lamina be not parallel to the fissure, UV may be conceived to be a physical line in the lamina, very distant from and parallel to the line AB, previously to the formation of the fissure, since the position or rectilinearity of such a line will not be sensibly affected by the opening of the fissure, as appears from the text.

extremity, become evanescent; but since the line is extended, though not by a force at its extremity P, it must at every other point be subjected to a certain tension, and our object will be to compare this tension at any point Q with that acting in the direction EQF at the same point, with the view of ascertaining within what limits another fissure might be formed subsequently to the formation of AB, and parallel to it between the lines AU and BV. Such a fissure could not be formed through Q, by the tensions to which we are supposing the lamina subjected, if the tension in the direction EQF at that point should be greater than that in the direction PS, since the fissure must necessarily be formed perpendicular to the greater of these tensions (Art. 6).

24. In the first place, let us suppose a physical line of indefinitely small width to be attached at its extremities to the fixed points A, B, and then conceive parallel forces to act on each element of this line,

B

with the same or different intensities at different points, and in directions perpendicular to AB. The line will thus be made to assume curvilinear form, and if the extensibility be small, as we shall suppose it to be, the curvature will be small, so that if AQ=s, and x be the original length of AQ, x and s may be considered as very approximately equal. Let denote the tension at Q, p the radius of curvature, and the intensity of the force at that point, being any function of x. Then the force on the element ds, will be p.8x, and the normal force produced by the tension 7, will estimated by the effect

T

=

it would produce, if it acted uniformly on a unit of the line, so that

T

the normal force acting on the element ds, will =

Sx very ρ

approximately. Consequently, if the normal make an angle AB, we shall have for the conditions of equilibrium of ds,

Again, let us suppose another physical line exactly similar and equal to the former, with its extremities fixed to two other points in lines through A and B respectively, and perpendicular to AB, and so that the two lines shall be in contact, when not acted on by any force. When the force acts in exactly the same manner on both, they will assume exactly similar positions, and those elements of the two lines respectively which were in contact when the lines were straight, will remain so when they have assumed their curvilinear form, and will be in exactly the same relative positions with respect to each other, as if the lines had been united into one previously to their becoming curved. Whence it follows, that there can be no more action between these lines when united, as we have just supposed, than if they were perfectly independent, and therefore the tension of each must remain the same as if this independence existed. If we conceive any number of lines to be united in a similar manner, so as to form a lamina, the same conclusion will apply to each.

25. Let us now take then a rectangular lamina ABGH, which we may conceive to be formed in this manner, and which we will suppose to be brought into the position represented in the annexed figure, by the force acting perpendicularly to AB, and in the plane of the lamina. EF represents a physical line originally parallel to AB; and PM another originally straight and parallel to AH, and therefore, still evidently remaining so, though in a different position, in the curved form of the lamina. Let x be the original distance of PM from AH,

be the original

dy

which will be approximately = AP, or EQ; then will & width of the element PM; and if AE, or PQ = y, Sy will be the width of EQF. Also, if T denote the tension of the lamina at Q, (estimated as in Art. 2.), in the direction of a tangent to EQF at

that point, it is evident that T. Sy will equal the tension above denoted Therefore the force produced by this tension in the direction of

T

by

T.

the normal to EQF at Q, will. Sa. Sy, acting on

ρ

.Sr. Sy, acting on the element

common to the two physical lines PM and EQF at Q.

T

will remain unaltered

P

Now it is manifest, that the tension T, and so long as the position of every element of the lamina remains so, whatever be the forces by which it is kept in that position. The action of will be the same at any point Q in PM as at P, since Q and P are similarly situated points in EQF and APB, and by hypothesis this force acts in the same manner upon each physical line, similar to APB. Consequently, the whole force on PM = 4. PM.8x. Let us suppose this force instead of acting on each element of PM, to be applied entirely at its extremity M. If this be done to every such line as PM, and the lamina be sensibly inextensible in the direction of these T lines, the position will remain undisturbed, and the normal force

Ρ

Sx. Sy,

at Q will not be altered. Hence, if T denote the tension of the lamina at Q in the direction PM, and therefore T'Sx the tension of PM at that point, and n the angle which the normal there to EQF makes with PM, we shall have for the conditions of equilibrium of the element common to PM and EQF,

26. If instead of supposing the lamina inextensible in the direction PM, we suppose it capable of small extension in that direction as well as in that parallel to AB, and still assume it to be acted on by forces applied at each point of HMG, so as to keep that extreme boundary in the same position as before, the physical line EF will assume a position differing in a small degree from its former one. Since the angle ʼn will still be very small, we shall still have T = const. nearly. The curvature at Q will no longer be the same as that at M, and ρ will therefore be a function of y, as well as of x. Consequently equation (1) of the